## $\int_{0}^{\pi} x f(sin x) dx$ is equal to

Q: $\displaystyle \int_{0}^{\pi} x f(sin x) dx$ is equal to

(A) $\frac{\pi}{2} \int_{0}^{\pi} f(sin x) dx$

(B) $\pi \int_{0}^{\pi/2} f(sin x) dx$

(C) $2 \pi \int_{0}^{\pi/2} f(sin x) dx$

(D) none of these

Ans: (A) , (B)

Sol: Let $\displaystyle I = \int_{0}^{\pi} x f(sin x) dx$

$\displaystyle = \int_{0}^{\pi} (\pi – x ) f sin (\pi-x) dx$

$\displaystyle I = \pi \int_{0}^{\pi} f(sin x) dx – I$

$\displaystyle 2 I = \pi \int_{0}^{\pi} f(sin x) dx$

$\displaystyle I = \frac{\pi}{2} \int_{0}^{\pi} f(sin x) dx$

$\displaystyle I = 2 \times \frac{\pi}{2} \int_{0}^{\pi/2} f(sin x) dx$

$\displaystyle I = \pi \int_{0}^{\pi/2} f(sin x) dx$

Hence (A) and (B) are the correct answers.